
This is a decoupled system of BSDE.
From Theorem 4.1 and Theorem 5.2 of [1]}, the \(i^{\mbox{th}}\)  equation in (REF ) admits a unique, hence maximal solution (see page 565 of [2]} for its definition) \((\underline{P}_{1,N}(i), \ \underline{\Lambda }_{1,N}(i))\in L^\infty _{\mathcal {F}^W}(0, N; \mathbb {R})\times L^{2}_{\mathcal {F}^{W}}(0, N;\mathbb {R}^{n})\) , and \(\underline{P}_{1,N}(i)\ge 0\)  for all \(i\in \mathcal {M}\) . From the proof of Theorem 3.5 of [3]}, the solution \((P_{1,N}(i), \ \Lambda _{1,N}(i))_{i\in \mathcal {M}}\)  of (REF ) could be approximated by the solutions of a sequence of BSDEs with Lipschitz generators. Thus we can use comparison theorem for multidimensional BSDEs (see for example Lemma 3.4 of [3]}) and then pass to the limit to get \(P_{1,N}(i)\ge \underline{P}_{1,N}(i)\)  for all \(i\in \mathcal {M}\) .
